6 research outputs found
Classification of -dimensional metric -Lie algebras
In this paper, we focus on -dimensional metric -Lie algebras. To
begin with, we give some properties on -dimensional -Lie algebras.
Then based on the properties, we obtain the classification of
-dimensional metric -Lie algebras
The extremal unicyclic graphs of the revised edge Szeged index with given diameter
Let be a connected graph. The revised edge Szeged index of is defined
as , where
(resp., ) is the number of edges whose distance to
vertex (resp., ) is smaller than the distance to vertex (resp.,
), and is the number of edges equidistant from both ends of
, respectively. In this paper, the graphs with minimum revised edge Szeged
index among all the unicyclic graphs with given diameter are characterized.Comment: arXiv admin note: text overlap with arXiv:1805.0657
No mixed graph with the nullity
A mixed graph is obtained from a simple undirected graph ,
the underlying graph of , by orienting some edges of . Let
be the cyclomatic number of with
the number of connected components of , be the matching number of
, and be the nullity of . Chen et al.
(2018)\cite{LSC} and Tian et al. (2018)\cite{TFL} proved independently that
,
respectively, and they characterized the mixed graphs with nullity attaining
the upper bound and the lower bound. In this paper, we prove that there is no
mixed graph with nullity . Moreover,
for fixed , there are infinitely many connected mixed graphs with nullity
is proved
On the extremal cacti with minimum Sombor index
Let be a graph with edge set . The Sombor index and the reduced Sombor index of a graph are defined as and , respectively. Where and are the degrees of the vertices and in , respectively. A cactus is a connected graph in which any two cycles have at most one common vertex. Let be the class of cacti of order with cycles. In this paper, the lower bound for the Sombor index of the cacti in is obtained and the corresponding extremal cacti are characterized when and . Moreover, the lower bound of the reduced Sombor index of cacti is obtained by similar approach
The extremal unicyclic graphs with given diameter and minimum edge revised Szeged index
Let be a connected graph. The edge revised Szeged index of is defined as , where (resp., ) is the number of edges whose distance to vertex (resp., ) is smaller than to vertex (resp., ), and is the number of edges equidistant from and . In this paper, the extremal unicyclic graphs with given diameter and minimum edge revised Szeged index are characterized